Let me point out at least that it is relatively consistent with ZFC that there is such a group. Indeed, it is a consequence of Martin's Axiom plus $\neg$CH.
This is because any dense subgroup of size less than the additivity $\mathfrak{a}$ of the null ideal will have all non-measurable selectors (see this MO question for more information about what this means). This follows by essentially the same reasons as for the Vitali set. Namely, suppose that $\Gamma$ is a dense subgroup of $\mathbb{R}$ of size less than $\mathfrak{a}$, and suppose that $V$ selects one element from each equivalence class by $\Gamma$-translation. Thus, $\mathbb{R}$ is the disjoint union of $|\Gamma|$ many translations of $V$. So $V$ cannot have measure $0$, since by assumption the union of $|\Gamma|$ many measure zero sets still has measure zero, and $V$ cannot have positive measure, since then it will have positive measure on an finite interval, and one can proceed just as in the Vitali case, finding infinitely many disjoint positive measure sets in a bounded interval, contradiction.
Finally, it is known to be consistent with ZFC that the additivity number can be strictly larger than $\omega_1$. This is a consequence, for example, of Martin's Axiom plus $\neg$CH.